EXPECTED RETURN, RISK, AND DIVERSIFICATION As managers, we rarely are consider investing in only one project at

EXPECTED RETURN, RISK, AND DIVERSIFICATION As managers, we rarely are consider investing in only one project at

time. Small businesses and large corporations alike can be viewed as collections of different investments, made at different points in time. We refer to a collection of investments as a portfolio.

While we usually think of a portfolio as a collection of securities (stocks and bonds), we can think of a business in much the same way—

a portfolio of assets such as buildings, inventories, trademarks, patents, and so forth. As managers, we are concerned about the overall risk of the business’s portfolio of assets.

Suppose you invested in two assets, Thing One and Thing Two, hav- ing 20% and 8% returns over the next year. Suppose you invest equal amounts, say $10,000, in each asset for one year. At the end of the year you expect to have $10,000(1 + 0.20) = $12,000 from Thing One and $10,000(1 + 0.08) = $10,800 from Thing Two, or a total value of $22,800 from our original $20,000 investment. The return on our portfolio is therefore:

------------------------------------------------- $22,800 $20,000 Return – = = 14% $20,000

If instead, we invested $5,000 in Thing One and $15,000 in Thing Two, the value of our investment at the end of the year would be:

THE FUNDAMENTALS OF VALUATION

Value of investment = $5,000 1 ( + 0.20 ) $15,000 1 0.08 + ( + ) = $6,000 + $16,200 = $22,200

and the return on our portfolio would be:

Return – = ------------------------------------------------------------------------------------------------------------------------------- ( ) $15,000 1 0.08 ( + ) $20,000 = 11%

which we can also write as: $5,000

Return = --------------------- ( 0.20 ) + --------------------- ( 0.08 ) = 11% $20,000

As you can see more immediately by the second calculation, the return on our portfolio is the weighted average of the returns on the assets in the portfolio, where the weights are the proportion invested in each asset.

We can generalize the formula for a portfolio return, r p , as the weighted average of the returns of all assets in the portfolio, letting:

a particular asset in the portfolio w i = proportion invested in asset i r i = return on asset i S = number of assets in the portfolio

Thus, . r p = w 1 r 1 + w 2 r 2 + …w + S r S We can write more compactly as:

∑ (10-8) i r i

Diversification and Risk In any portfolio, one investment may do well while another does poorly.

The projects’ cash flows may be “out of sync” with one another. Let’s see how this might happen.

Suppose you own Asset P that produces the returns over time shown in Exhibit 10.4(a). These returns vary up and down within a wide range. Suppose you also invested in Asset Q whose returns over time are shown in Exhibit 10.4(b). These returns also vary over time within a wide band. But since the returns on Asset P and Asset Q are out of sync, each tends to provide returns when the other doesn’t. The result is that your portfo- lio’s returns vary within a narrower range as shown in Exhibit 10.4(c).

Risk and Expected Return

EXHIBIT 10.4 Returns on Asset P, Asset Q, and a Portfolio over Time

Panel A: Returns on Asset P over Time

Panel B: Returns on Asset Q over Time

Panel C: Returns on a Portfolio Comprised of Asset P and Asset Q over Time

THE FUNDAMENTALS OF VALUATION

Let’s look at the idea of “out-of-syncness” in terms of expected returns, since this is what we face when we make financial decisions. Con- sider Investment C and Investment D and their probability distributions:

Return on Scenario

Probability of

Return on

Scenario Investment C Investment D

We see that when Investment C does well, in the boom scenario, Investment D does poorly. Also, when Investment C does poorly, as in the recession scenario, Investment D does well. In other words, these investments are out of sync with one another.

Now let’s look at how their “out-of-syncness” affects the risk of the portfolio of C and D. Suppose we invest an equal amount in C and D. The calculation of the expected return and standard deviation for Investment C, Investment D, and the portfolio consisting of C and D is shown in Exhibit 10.5. The expected return on Investment C is 2% and the expected return on Investment D is 6%. The return on a portfolio comprised of equal investments of C and D is expected to be 4%. The standard deviation of Investment C’s return is 14% and of Investment D’s return is 19.97%, but the portfolio’s standard deviation, calculated using the weighted average of the returns on Investment C and D in each scenario, is 4.77%. This is less than the standard deviations of each of the individual investments because the returns of the two invest- ments do not move in the same direction at the same time, but rather tend to move in opposite directions.

The portfolio comprised of Investments C and D has less risk than the individual investments because each moves in different directions with respect to the other. A statistical measure of how two variables—in this case, the returns on two different investments—move together is the covariance. Covariance is a statistical measure of how one variable changes in relation to changes in another variable. Covariance in this example is calculated in four steps:

Step 1: For each scenario and investment, subtract the investment’s expected value from its possible outcome. Step 2: For each scenario, multiply the deviations for the two invest- ments. Step 3: Weight this product by the scenario’s probability. Step 4: Sum these weighted products to arrive at the covariance.

EXHIBIT 10.5 Calculation of the Standard Deviations for Investment C, Investment D, and the Portfolio Comprised of Investment C and

Investment D Investment C

Probability

Deviation

Squared Deviation

Weighted

Squared Probability Return

Deviations Scenario

Return

Expected Value

Expected Value